Motion control systems are used in number of positioning applications, e.g., single-axis positioning, and multiple-axis positioning. For example, a simple single-axis positioning motion control system generally includes sensors, controller, amplifier, and actuator motor. The actuator follows a predetermined trajectory subject to state and control constraints, i.e., dynamics, acceleration, velocity. The trajectory of the actuator can be designed to reduce vibration induced by the motor.
For two motor control cases, FIGS. 1A-1C and 2A-2C show optimal prior art time profiles for position, velocity, and control input, respectively. In the first case, an acceleration constraint is always active, while in the second case, the velocity constraint is saturated in a coasting part of the velocity profile, and the acceleration constraints are active in the other parts of the velocity profile. It is clear that when the control is optimized for minimize time, the control input contains significant large transitions, which are energy inefficient.
Although minimal time motor controllers generate the fastest trajectory for each motion, for a complex processes, minimal time controller may not help improve the overall productivity if a bottleneck of production is due to other slower processes, such as material processing. For example, there is no advantage in rapidly moving a work piece to a next state using excessive energy, if the piece is not going to be manipulated until later.
For such systems, minimal time controllers are not only unnecessary, but also inefficient because the controllers are not energy optimal. Furthermore, the efficiency of a plant depends not only on productivity, but also on other costs, such as energy consumption. The maximum efficiency is usually generated with certain trade-off between productivity and energy consumption. Therefore, strictly minimal time controllers, although useful in certain cases, do not increase efficiency in general, and minimizing energy consumption by relaxing time constraints should be considered for optimal motor control.
Optimal Control Theory
Optimal control deals with the problem of finding a control law for a system such that a certain optimality criterion is achieved. The control problem includes a cost function of state and control variables. An optimal control has to satisfy a set of differential equations describing paths of the control variables that minimize the cost function.
Pontryagin's minimal principle for optimal control theory determines the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints on the state or control inputs. The optimal control theory provides a systematic way for determining the optimal solution to the problem of minimizing certain cost functions, such as time and energy, subject to various constraints, including dynamics constraint, boundary conditions (BC), state constraint, control constraint, and path constraint. Therefore, the energy efficient motor control problem can be addressed as an optimal control problem.
The optimal control can be obtained by solving a two-point boundary value problem (TBVP), or a multi-point boundary value problem (MBVP) if the optimal solution contains multiple segments. This usually happens when control or state constraints are active. For the minimal time motor control problem, the optimal solution can be obtained analytically. Such an analytic solution forms the basis of many minimal time motor controllers.
However, for an energy saving optimal control problem, the corresponding TBVP and MBVP are difficult to solve, and no analytic solution is readily available. The existing indirect methods for solving the TBVP and MBVP, including single shooting method (SSM) and multiple shooting method (MSM), are computationally complex for real-time motion control applications. Besides, the convergence of those methods are generally not guaranteed, and rely on an initial guess of certain key parameters in the methods. Hence, due to the computation complexity issue and the reliability issue, the existing methods for solving TBVP and MBVP are difficult to be applied for real-time energy efficient trajectory generation in motor control applications.
The direct transcription method (direct method), provides an alternative way for solving optimal control problems. Similar to shooting methods, the convergence of the direct method is not guaranteed. A comprehensive evaluation of current direct methods, including a pseudo-spectral method and a mesh refinement method, shows that the direct method cannot provide motor control in real-time.
Thus, the known methods are insufficient in terms of computation efficiency and reliability for the real-time application of energy saving motor control. Due to these difficulties, there is a need for a method to generate energy efficient reference trajectories for motor control. Such a method should be computationally efficient for real-time motor control applications, and should be reliable. It is also desirable that such a method provides the capability to adjust the trade-off between execution time and energy saving for different applications.